The Self-Excitation Damping Ratio in Variable Speed Milling

Transcription

1 Proceedings of the World Congress on Engineering 01 Vol III, Jul 4-6, 01, London, U.K. The Self-Excitation Damping Ratio in Variable Speed Milling K. Saleh and N.D. Sims Abstract During High Speed Machining (HSM), unstable self-excited vibrations (known as regenerative chatter) can occur, leading to poor surface finish, excessive tool wear, and damage to the machine. Consequentl there has been a great deal of research to understand the mechanisms of regenerative chatter so that productivit can be enhanced. A chatter suppression method that has received attention recentl is the spindle speed variation technique. However, this approach results in stabilit boundaries that are defined b dela-differential equations with periodic coefficients and variable time delas. In the present stud, a Simulink model which is designed for milling chatter simulation is modified to enable its use under variable spindle speed circumstances. This is achieved b rewriting the equations of motion in nondimensional time. A novel signal processing technique is then adopted for analsing the chatter stabilit. Kewords: Milling Speed Variation, Stabilit in HSM, Signal Processing, Time Domain Simulation. I. INTRODUCTION M illing with high speed machining is one of the common processes used in producing a variet of the industrial products. However, the productivit and qualit is limited due the vibrations that arise during the cutting process. These vibrations cause poor surface finish, increase the rate of tool wear and reduce the tool lifetime. Self-exited vibration (chatter) induced b regeneration of surface waviness is the one of main reasons for such vibrations. One solution that has received some attention is the use of continuousl varing spindle speeds during milling [1]. In general, models designed for determining and analzing machining stabilit of Constant Speed Machining (CSM) are not directl applicable to Variable Speed Machining (VSM) due to the presence of the time varing dela in the differential equations [7]. Therefore some studies (for example [8-10]) have presented different mathematical techniques to be used for modeling the delaed differential equation with a time varing dela term. In addition a number of the time domain simulations and analtical approaches have been developed for chatter detection. However the problem arises of how to ascertain, based upon the computed data, whether the simulated cut was stable or unstable. The Peak-To-Peak (PTP) technique K. Saleh is with Department of Mechanical Engineering, The Universit of Sheffield, Mappin Street, Sheffield S1 3DJ, UK (Tel: ; Fax: ; mep08ksk@sheffield.ac.uk). N.D. Sims is with Department of Mechanical Engineering, The Universit of Sheffield, Mappin Street, Sheffield S1 3DJ, UK, (Tel: ; Fax: ; n.sims@sheffield.ac.uk) was used to identif the stabilit boundar during the CSM as in [9, 11-1]. However for VSM case, it can be seen that the sstem displacement does not reach a stead state condition, making the PTP method difficult to use as a judgment of chatter stabilit. Therefore a signal processing method is considered as an effective method used to detect chatter and the stabilit boundar for this case. Sims [11], implemented this approach for CSM case. The structure of this paper as follows: It begins with the representation for time domain milling in CSM and VSM. A signal processing method is then described. Results are then presented starting with comparing the stabilit of the VSM versus the CSM case. Then results from changing the acceleration of the spindle speed linearl and b taking the triangular variation waveform are analsed. Finall the overall results of this stud are concluded. II. TIME DOMAIN MILLING WITH CSM Consider the schematic representation of milling shown in Figure 1. Here, a milling tool is removing material from a workpiece. For simplicit the tool is shown with a single degree of freedom in the -direction, although the following analsis can be readil extended to the multi-degree-offreedom case. In order to predict the time response of this sstem, a discretised model is formulated in the Simulink modelling environment as shown in Figure. The model consists of three aspects, namel milling kinematics, milling forces, and sstem dnamics. These components will now be briefl summarised more detailed descriptions of the modelling procedure can be found in references [11, 13]. The kinematics model begins b dividing the tool into discrete axial slices (such as that shown in Figure 1, and calculating the tool and workpiece geometr within each slice. Two coordinate sstems are used: a radial coordinate based upon the centre of the tool with angles taken relative to the feed direction, and a Cartesian coordinate sstem based upon the workpiece feed direction. The relative displacement of the workpiece and tool (accounting for feed rate and vibration effects) are provided as inputs to the calculation. The basis of the computation is the manipulation of a set of arras of Cartesian coordinates (one arra for each tooth on each axial slice), that define the surface of the workpiece that was produced b that tooth. The arra length represents a complete revolution of the tooth, therefore with each tooth revolution the arra values are overwritten, or updated. For each time step in the simulation, the following calculations are repeated for each tooth on each axial slice: 1) The position of the tooth is calculated based upon the current simulation time, and the spindle speed.

2 Proceedings of the World Congress on Engineering 01 Vol III, Jul 4-6, 01, London, U.K. ) The workpiece surface arra for the present tooth is updated. This surface arra is depicted b the o markers on the workpiece surface in Figure 1. 3) The instantaneous chip thickness h t for the present tooth is calculated, based upon the current tooth position, and the surface arra for the preceding tooth. This kinematic model was programmed in the C programming language, and interfaced with the Simulink model via a mex-function. The milling forces computation takes the instantaneous chip thicknesses that have been calculated for each tooth N t and axial slice l of the kinematics computation. The corresponding cutting forces are determined as a radial component F r and a tangential component F t opposing the rotation of the tool, and are assumed proportional to the instantaneous chip thickness [7]. The forces are then transformed into the workpiece Cartesian coordinates, and numericall integrated to give the resultant cutting forces in the x and -directions. The sstem dnamics are then modeled in standard transfer-function formulations, so as to calculate values for the vibration in the x and directions (as appropriate). tool ω k c θ u N t l Workpiece rt x N t-1 F r F t l -1 feed Figure 1 One-Dimensional Model of Milling Process Figure Simulink model of milling vibrations. III. TIME DOMAIN MILLIN WITH VSM In the present work, the time-domain model shown in Figure will be modified for use in variable speed simulateions. A detailed description of this model can be found in [11, 13]. Traditionall real-time t is used as the independent variable for writing the solution of the equations of motion. This is particularl the case when simulating dnamic sstems using the Simulink modelling environment. Here, phsical time (with units of seconds) is assumed to be the independent variable, and a wide range of numerical integration routines are available for solving the equations of motion (e.g. the 4 th order Runge Kutta method, etc). However, a close inspection of the Simulink formulation described above reveals that the milling kinematics model requires a fixed number of time steps for each revolution of the tool. This means that a fixed-step solver (e.g. the Runga Kutta method) must be used, and the spindle speed must be fixed. In this section, this problem is overcome b using tool revolution as the independent variable in rewriting the sstem equation of motion. Consequentl, Simulink time is no longer equal to phsical time, but rather the number of tool revolutions. With this approach, a fixed step solver will alwas involve a fixed number of time steps per tool revolution, even if the spindle speed is changed. This concept will now be derived more formall. With reference to Figure 1, the single degree of freedom sstem representing the tool vibration can be described b: d d m c () k t F (1) dt dt It should be re-iterated that the force F is a function of the current and delaed vibration, along with the instantaneous angle u of the tool. Consequentl, equation (1) is a dela differential equation with nonlinear and time periodic coefficients. Loss of stabilit could be associated with secondar Hopf bifurcations, cclic fold bifurcations, or period doubling bifurcations, which in practice are collectivel referred to as regenerative chatter. The relationship between phsical time t and instantaneous spindle speed is: d (rev/sec) () dt Where τ is the number of tool revolutions, the velocit can then be rewritten using the chain rule as: d d d d dt d dt d (3) Moreover the sstem acceleration can be rewritten as: d d d d dt d d dt (4) Here, the rate of change of spindle speed can be rewritten to give: d d d d (5) dt d d d The equation of motion (1) can then be written in terms of the derivative expressions on the right-hand-side of equations (3) and (5).

3 Proceedings of the World Congress on Engineering 01 Vol III, Jul 4-6, 01, London, U.K. d d d d F k () c m (6) d d d d This can be rearranged to give: d F k ( ) c d 1 d d (7) d m m m d d d The sstem dnamics which are represented b the equation of motion (7) give rise to the Simulink sstem shown in Figure 3. A Time Domain Simulation B Divided signal into F frames each of length N, representing 1 tool ccle X (n,f) C Take the discrete Fourier transform X (n,f) Simulation Parameters (depth of cut, spindle speed, cutting geometr, tool geometr and simulation settings E Find spectral lines k p which have maximum values F Take Logarithms to show the self excited vibration behaviour per ccle (i.e. frame) D Set tooth passing harmonics to zero X se (m*n,f)=0, m=1,,3.. G Fit a 1st order polnomial and select the spectral line k p,max With the steepest gradient Figure 4 Flow chart to illustrate evaluation of the chatter criterion ζ Figure 3 Simulink model structure in non-dimensional time IV. SIGNAL PROCESSING METHOD In metal cutting, self-excited vibrations that cause chatter are not clearl evident, since the forced vibrations (due to the rotation of the tool) are dominating the response of the sstem. This is particularl the case in the region of marginal chatter instabilit, where the self-excited vibration will grow ver slowl. Therefore, man chatter criteria require the simulation to run for a long period of time before it is possible to determine if the sstem is stable or unstable. Consequentl the technique of isolating the self-excited vibration from the forced vibration was adopted particularl in milling machining. The procedure of this technique is now briefl summarized in Figure 4 and more description can be found in [11]. The first step (block A) is a time-domain simulation of the cutting process under given cutting parameters. These data signals are then divided into number of frames, each frame represents one complete tool ccle (block B). A Fourier analsis approach is then applied to each of these data sets (block C). Isolating the self-excited vibration from the forced vibration is then executed (block D). Maximum values of the spectral lines are then determined (block E). These lines can then be plotted on a natural logarithmic scale, thereb indicating the self-excited vibration behavior per revolution of the tool (Block F). Finall (block G) is the step of calculating the self-excited damping ratio ζ b fitting a first order polnomial and selecting the spectral line (k p,max ) with the steepest gradient. V. STABILITY ANALYSIS OF VSM In the present contribution, the time domain modeling approach is combined with the signal processing method in order to illustrate the stabilit of the VSM. A simple milling scenario with one degree of freedom is considered. The techniques and the procedures that are used in the current model can be applied to sstems with multiple modes of vibration, and involving interrupted cutting with multiple teeth engaged in the cut simultaneousl. It is also straightforward to include a nonlinear relationship between cutting force and chip thickness, and the periodic excitation force is inherentl modeled. Model validation is then presented b repeating some of the analsis presented in [4, 9]. The milling parameters were also chosen to closel match those used in previous work b [4, 9] and are summarized in Table 1. A variable spindle speed was then used in the time-domain model, with a triangular speed variation around a mean value. This was defined in accordance with reference [4, 9], with fixed parameters of RVA and RVF. To begin with demonstrating the difference in the chatter behavior for CSM and VSM, a small selection of the results is considered. In addition the stabilit lobe diagram for the constant spindle speed case is produced. The self- excited damping ratio technique is then used for stabilit investigation. Here chatter was analsed b considering a two cases of the milling speed variation. In the first case, and at a constant depth of cut of 1mm, milling speed was accelerated from 7500 to rpm. This procedure was performed for different scenarios of the spindle accelerations. The second case, milling speed was assumed to be a variable, with a periodic triangular speed variation around a mean value m =9100 rpm. For the timedomain simulations, the signal processing technique is used to analze the data signal of the simulated chatter and to determine so called self-excited damping ratio of each case of the simulation. It should be pointed out that the parameters are chosen in the present stud ma not completel match those used in reference [4, 9] since the tool helix angle; tool radial immersion; number of tool ccles, and number of simulation axial laers were not given.

4 VSS (rpm) (m) (m) (m) Proceedings of the World Congress on Engineering 01 Vol III, Jul 4-6, 01, London, U.K. Table I Input Simulation Parameters Sample Parameter Value D Tool diameter 5 mm N t Number of teeth 3 Flute helix 0 0 r imm Radial immersion mm f pt Feed per tooth 1 mm m Mass kg f n Natural Frequenc.5 Hz ζ Damping ratio K t and K r Tangential and Radial 700 and cutting stiffness 140 MPa RVA Amplitude ratio 0.8 RVF Frequenc ratio mm=millimetre, kg=kilogram, Hz=Hertz, MPa= Mega-Pascal 0 x x 10-7 x 10 5 (a) Simulation Results of CSM (=9100 rpm and b=1mm) (b) Simulation Results of VSM (=9100 rpm and b=1mm) (c) Compare displacement behavior CSM vs VSM VI. RESULTS A. Stabilit Analsis for CSM versus VSM Before presenting an analsis of the chatter stabilit using the self-excited damping ratio method, it is worth illustrating the time response of the sstem, and demonstrating how continuousl varing the spindle speed can suppress the chatter level more than the constant case. Here the CSM and VSM simulation results for last 30 ccles are compared at the same simulation conditions. Figure 5 presents the simulation results of the sstem displacement for both CSM and VSM with the depth of cut (b=1mm) and the constant and mean value of the spindle speed ( m =9100 rpm). With reference to [4, 9], this is a small selection just to show the effectiveness of the VSS technique in suppressing the chatter levels. A tpical result of the case of constant speed machining CSM (=9100 rpm) is shown in plot (a), here the chatter level is clearl growing, and consequentl the sstem is unstable. However plots (b) and (d) demonstrate the case of variable speed machining. Here, the triangular waveform of the spindle speed is varing periodicall about the mean value of the speed ( m =9100 rpm) shown in plot (d). Plot (b) illustrates the overall chatter level which is clearl stabilized at a level significantl lower compared to the CSM case (plot (a)). Thereb the sstem here is considered to be stable. The comparison between the CSM and VSM is further illustrated in plot (c). From this plot it can be clearl noticed that VSM approach has lower and almost constant vibration which indicates significantl greater stabilit than for the constant spindle speed condition. Therefore these results agree reasonabl well with those given in [4, 9], and the small differences could be attributed to the different parameters used in the simulations. Consequentl these outcomes have validated the proposed variable spindle speed modeling procedure. However, close inspection of the displacement signal shown in Figure 5 reveals that the response of the transient behavior of the signal is periodicall arising which can be attributed to the spindle speed alterations. This makes it difficult to appl standard approaches for fairl analzing the stabilit. Here the selfexcited damping ratio approach is considered to be the more appropriate method for analzing milling stabilit behavior (d) Triangular Speed Variation about the mean value of VSM ( m =9100 rpm) Time (sec) Figure 5 Simulation results for CSM and VSM B. Stabilit Analsis for Accelerating Spindle Speed As can be seen in section A chatter levels are significantl reduced when the spindle speed varies periodicall with triangular waveform. Now in this section, effects of linearl varing the spindle speed with different accelerat-ions behaviors will be investigated. Figure 6 shows the overall results of changing the spindle acceleration on the milling stabilit behavior. Plot (a) shows the stabilit lobe with markers (*) A and ( ) B are corresponding to the start and end of the milling speed variation. In this case, simulation starts from an unstable region at 7500 rpm (associated with a secondar Hopf bifurcation) and ends b the stable region at rpm, while markers at ( ) C, ( ) D and ( ) E correspond to the boundar of the stabilit at that defined milling speeds at the same depth of cut 1mm. Plot (b) shows the variation of the milling speed from 7500 to rpm with accelerations 1.5, 5, 50 and 15 rpm/rev, while plots (c), (d), (e) and (f) of (Figure 6) show the evaluation of the chatter obtained b the time domain simulation, at these variable milling speeds and the depth of cut 1mm. As can be seen from the plot (c) the stages of the cutting stabilit are changing along the milling speed variation with a low acceleration of 1.5 rpm/rev. Here milling machining starts from unstable region where the chatter grows steadil between points A and C, this region is associated with a secondar Hopf bifurcation. Then the signal of the self-excited vibration is decaing between the points C to D and beond the point E, these are corresponding to stable cutting regions. However chatter vibrations are steepl growing between points D (star) and E. This region corresponds to the unstable cut along with the flip bifurcation region.

5 SSV (rpm) SSV (rpm) b (mm) Proceedings of the World Congress on Engineering 01 Vol III, Jul 4-6, 01, London, U.K (a) Stabilit Lobe Diagram for CSM Range of the TSV A C D E B Spindle Speed (rpm) (b) Ramped Spindle Speed from 7500 to rpm C. Stabilit Analsis for Triangular Speed Variation For the same range of the spindle speed variation that is highlighted on plot (a) of (Figure 6), the depth of cut was now changed, and the chatter is analzed as shown in Figure 7. Plot (a) shows the triangular speed variation around the mean value m =9100 rpm in a periodic fashion, with a fundamental period 6 tool revolutions per one tool ccle, while plot (b) shows the results of the chatter analsis obtained from simulating this speed with the given frequenc ratio RVF and amplitude ratio RVA. Accel=1.5 rpm/rev Accel=5 rpm/rev Accel=50 rpm/rev Accel=15 rpm/rev (c) Chatter Analsis for Spindle Acceleration 1.5 rpm/rev (d) Chatter Analsis for Spindle Acceleration 5 rpm/rev (e) Chatter Analsis for Spindle Acceleration 50 rpm/rev (f) Chatter Analsis for Spindle Acceleration 15 rpm/rev Tool Revolution (revs) Figure 6 Effect of the changing speed acceleration in milling stabilit. However the stable region between points D and D in plot (c) is inspected and can be attributed to the acceleration of the spindle. Plots (d), (e) and (f) demonstrate the stages of the chatter behavior obtained from accelerating the spindle speed faster at 5, 50 and 15 rpm/rev respectivel. Here the result obtained from the time domain simulation is clearl demonstrating that, stabilit region (C to D) was reduced b increasing the milling speed acceleration through all the cases. However there is a positive point of this approach which is that the sharp growth of the chatter (D to E) through the flip bifurcation region is reduced b increasing the spindle acceleration. This result is considered as worth of more investigations for high speed machining. D' 11, (a) Periodic Triangular Speed Variation (TSV) ( m =9100 rpm) Tool Revolution (revs) (b) Chatter Analsis of the Periodic TSV docb=3mm = docb=.6mm = docb=mm = docb=0.8mm = docb=0.mm = Frame Number (3f : ) Figure 7 Model simulation results, (a) Triangular milling speed variation, (b) chatter analsis due to the periodic milling speed. In this section, the analsis frames have been overlapped to three frames per two ccles of the tool revolutions (3f:τ), so as to increase the number of data points available. As can be seen from the plot (b) for depth of cuts 0. and 0.8 mm the data of the self-excited vibrations (fitted b straight lines) declining with each tool rotation, and the patterns of the signal are in a periodic fashion, giving a positive damping ratios of ζ= and.0003 respectivel. Increasing depth of cut results is decreasing of the damping ratio values (ζ 0). This indicates that the sstem is approaching the stabilit boundar where the value of the ζ= 0. This can be clearl realized from the case of the depth of cut mm, here the signal of the chatter is almost periodic with less slope inclination of the vibration data, giving a positive damping ratio value ζ= This means that at this depth of cut the sstem is approximatel reaching the stabilit boundar, which this case gives a more stabilit than the CSM case. However increasing the depth of cut to.6 mm the sstem becomes unstable. The self-excited vibrations now grow with each ccle, and the signal is start losing its periodicit, giving a negative ζ = Increasing depth of cut 3mm causes the instabilit to worsen, the vibrations grow rapidl at a higher rate, and the signal is clearl losing the periodicit behavior, giving ζ = Similar results were acquired b using PTP method in [4, 9], and the small differences could be attributed to the different parameters used in the simulations. Consequentl

6 Proceedings of the World Congress on Engineering 01 Vol III, Jul 4-6, 01, London, U.K. these outcomes have validated the proposed variable spindle speed modeling procedure. VII. CONCLUSION In this stud, a comprehensive milling simulation model has been modified to account for milling scenarios that involve a variable spindle speed. The model is formulated in a Simulink environment, and in order to accommodate variable spindle speeds the sstem equation of motion has been reformulated in non-dimensional time. As a result, the simulation time used as the independent variable in Simulink becomes the tool revolution, rather than the simulated time in seconds. The model results have been validated b comparing a small selection of simulation results with the work presented in reference [4, 9]. Reasonable agreement was observed, but the present stud has re-enforced the issue of analzing chatter stabilit for variable spindle speed simulations. Chatter amplitude can be significantl reduced b using a periodic triangular speed variation approach. In addition, accelerating the spindle speed linearl shows some positive results of reducing the sharp growth of the chatter through the flip bifurcation region. This promise is considered as worth of more investigations for high speed machining. It is clear that the signal processing approach is able to provide a formal interpretation of chatter stabilit for variable spindle speed simulations. ACKNOWLEDGMENT The authors are grateful for the support of the EPSRC through grant reference EP/D05696/1. [3] Pakdemirli, M. And A.G. Ulso, Perturbation Analsis Of Spindle Speed Vibration In Machine Tool Chatter. Jvc/Journal of Vibration and Control, (3): P [4] Sébastien Segu, G.D., Lionel Arnaud, Tamás Insperger, On the Stabilit of High-Speed Milling with Spindle Speed Variation. The International Journal of Advanced Manufacturing Technolog, 009: P [5] Sri, N.N. And R. Beddini, Spindle Speed Variation for the Suppression of Regenerative Chatter. Journal of Nonlinear Science, (3): P [6] Yilmaz, A., Et Al., Machine Tool Chatter Suppression b Multi-Level Random Spindle Speed Variation. Journal of Manufacturing Science and Engineering-Transactions of the Asme, (): P [7] Jaaram, S., Et Al., Analtical Stabilit Analsis Of Variable Spindle Speed Machining. Journal of Manufacturing Science And Engineering-Transactions Of The Asme, (3): P [8] Insperger, T. And G. Stepan, Stabilit Analsis of Turning with Periodic Spindle Speed Modulation via Semidiscretization. Journal of Vibration and Control, (1): P [9] S. Segu, T.I., L. Arnaud, G. Dessein, G. Peigné, Chatter Suppression In Milling Processes Using Periodic Spindle Speed Variation, In 1th Cirp Conference On Modelling Of Machining Operations. 009: Donostia-San Sebastián, Spain. [10] Tsao, T.C., Et Al., A New Approach to Stabilit Analsis of Variable- Speed Machining Sstems. International Journal of Machine Tools & Manufacture, (6): P [11] Sims, N.D., the Self-Excitation Damping Ratio: A Chatter Criterion for Time-Domain Milling Simulations. Journal of Manufacturing Science and Engineering-Transactions of the Asme, (3): P [1] Smith, S. And J. Tlust, Update On High-Speed Milling Dnamics. Journal of Engineering for Industr-Transactions of the Asme, (): P [13] Sims, N.D., Et Al., Analtical Prediction of Chatter Stabilit for Variable Pitch and Variable Helix Milling Tools. Journal of Sound and Vibration, (3): P b f CF c F r F t F h t k k p,max l NCF NDF NSF TSV SSV m τ u NOMENCLATURE Depth of cut Frame numbers Cutting forces Sstem damping coefficient Radial cutting forces Tangential cutting forces Total cutting forces in -direction Instantaneous dnamic chip thickness Sstem stiffness coefficient Maximum spectral line Number of axial laers Natural logarithmic for maximum spectral line per frame, see [11] Normalized cutting forces Normalised damping forces Normalized stiffness forces Triangular speed variation Spindle Speed Values Sstem displacement in -direction Constant value of the spindle speed Mean value of the spindle speed Tool position angle Number of tool revolutions REFERENCES [1] Al-Regib, E., Et Al., Programming Spindle Speed Variation for Machine Tool Chatter Suppression. International Journal of Machine Tools & Manufacture, (1): P [] Ismail, F. And E.G. Kubica, Active Suppression of Chatter in Peripheral Milling.1. A Statistical Indicator to Evaluate the Spindle Speed Modulation Method. International Journal of Advanced Manufacturing Technolog, (5): P